Interval completion with few edges

Transcription

1 Interval completion with few edges Pinar Heggernes Christophe Paul Jan Arne Telle Yngve Villanger Abstract We present an algorithm with runtime O(k 2k n 3 m) for the following NP-complete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the long-standing open question [17, 6, 24, 13], first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) n O(1). The problem has applications in Physical Mapping of DNA [11] and in Profile Minimization for Sparse Matrix Computations [9, 25]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively. This work is supported by the Research Council of Norway Department of informatics, University of Bergen, Norway. s: yngvev@ii.uib.no, pinar@ii.uib.no, telle@ii.uib.no CNRS - School of Computer Science, McGill University, Canada - LIRMM, Université Montpellier II, France. paul@lirmm.fr 1

2 1 Introduction and motivation Interval graphs are the intersection graphs of intervals of the real line and have a wide range of applications [12]. Connected with interval graphs is the following problem: Given an arbitrary graph G, what is the minimum number of edges that must be added to G in order to obtain an interval graph? This problem is NP-hard [18, 8] and it arises in both Physical Mapping of DNA and Sparse Matrix Computations. In Physical Mapping of DNA a set of contiguous intervals of the DNA chain, called clones, are given together with experimental information on their pairwise overlaps. The goal is to build a map describing the relative position of the clones. In the presence of false negative errors, the problem of building a map with fewest errors is equivalent to finding the smallest edge set whose addition to the input graph will form an interval graph [11]. In Sparse Matrix Computations, one of the standard methods for reordering a matrix to get as few non-zero elements as possible during Gaussian elimination, is to permute the rows and columns of the matrix so that non-zero elements are gathered close to the main diagonal [9]. The profile of a matrix is the smallest number of entries that can be enveloped within off-diagonal non-zero elements of the matrix. Translated to graphs, the profile of a graph G is exactly the minimum number of edges in an interval supergraph of G [25]. In this paper, we present an algorithm with runtime O(k 2k n 3 m) for the k-interval Completion problem of deciding whether a graph on n vertices and m edges can be made into an interval graph by adding at most k edges. The k-interval Completion problem is thus FPT 1, which settles a long-standing open problem [17, 6, 24, 13]. An early paper (first appearance FOCS 94 [16]) in this line of research by Kaplan, Shamir and Tarjan [17] gives FPT algorithms for k- Chordal Completion, k-strongly Chordal Completion, and k-proper Interval Completion. In all these cases a bounded search tree algorithm suffices, that identifies a subgraph which is a witness of non-membership in the desired class of graphs, and branches recursively on all possible ways of adding an inclusion-minimal set of edges that gets rid of the witness. The existence of an FPT algorithm for solving k-interval Completion was left as an open problem by [17], with the following explanation for why a bounded search tree algorithm seemed unlikely: An arbitrarily large obstruction X could exist in a graph that is not interval but could be made interval with the addition of any one out of O( X ) edges. It is therefore surprising that our FPT algorithm for this problem is indeed a bounded search tree algorithm. Let us mention some related work. Ravi, Agrawal and Klein gave an O(log 2 n)-approximation algorithm for Minimum Interval Completion, subsequently improved to an O(log n log log n)- approximation by Even, Naor, Rao and Schieber [7] and finally to an O(log n)-approximation by Rao and Richa [22]. Heggernes, Suchan, Todinca and Villanger showed that an inclusionminimal interval completion can be found in polynomial time [15]. Kuo and Wang [20] gave an O(n ) algorithm for Minimum Interval Completion of a tree, subsequently improved to an O(n) algorithm by Díaz, Gibbons, Paterson and Torán [4]. Cai [2] proved that k-completion into any hereditary graph class having a finite set of forbidden subgraphs is FPT. Some researchers have been misled to think that this settled the complexity of k-interval Completion, however, interval graphs do not have a finite set of forbidden subgraphs [21]. Gutin, Szeider and Yeo [13] gave an FPT algorithm for deciding if a graph G has profile at most k + V (G), but the more 1 A parameterized problem with parameter k and input size n that can be solved by an algorithm having runtime f(k) n O(1) is called fixed parameter tractable (FPT). 2

3 natural parametrization of the profile problem is to ask if G has profile at most k + E(G), which is equivalent to the k-interval Completion problem on G. Our search tree algorithm for k-interval Completion circumvents the problem of large obstructions (witnesses) by first getting rid of all small witnesses, in particular witnesses for the existence of an asteroidal triple (AT) of vertices. Since a graph is an interval graph if and only if it is both chordal and AT- free [21], to complete into an interval graph we must destroy witnesses for non-chordality and witnesses for existence of an AT. Witnesses for non-chordality (chordless cycles of length > 3) must have size O(k) and do not present a problem. Likewise, if an AT is witnessed by an induced subgraph S of size O(k) it does not present a problem, as shown in Section 3 of the paper. The difficult case is when we have a chordal non-interval graph G with no AT having a small witness. For this case we introduce thick AT-witnesses in Section 4, consisting of an AT and all vertices on any chordless path between any two vertices of the AT avoiding the neighborhood of the third vertex of the AT. We define minimality for thick AT-witnesses, and show that in every minimal thick AT-witness one of the vertices of the AT is shallow, meaning that there is a short path from it to each of the other two vertices of the AT. For the difficult case of G being a chordal graph having no small AT-witnesses, we are able to compute a set of vertices C consisting of shallow vertices such that removing C from the graph gives an interval graph. Based on the cardinality of C we handle this case by branching in one of several different ways of getting rid of the minimal thick AT-witness corresponding to a vertex in C. In particular, in Section 5 we show that when no bounded branching is possible the instance has enough structure that the best solution is a completion computed in a greedy manner. The presented algorithm consists of 4 branching rules. 2 Preliminaries We work with simple and undirected graphs G = (V, E), with vertex set V (G) = V and edge set E(G) = E, and n = V, m = E. For X V, G[X] denotes the subgraph of G induced by the vertices in X. We will use G x to denote G[V \ {x}] for x V, and G S to denote G[V \ S] for S V. For neighborhoods, we use N G (x) = {y xy E}, and N G [x] = N G (x) {x}. For X V, N G [X] = x X N G[x] and N G (X) = N G [X] \ X. We will omit the subscript when the graph is clear from the context. A vertex set X is a clique if G[X] is complete, and a maximal clique if no superset of X is a clique. A vertex x is simplicial if N(x) is a clique. We will say that a path P = v 1, v 2,..., v p is between v 1 and v p, and we call it a v 1, v p -path. The length of P is p. We will use P v p and P + v p+1 to denote the paths v 1, v 2,..., v p 1 and v 1, v 2,..., v p, v p+1, respectively. We say that a path P avoids a vertex set S if P contains no vertex of S. A chord of a cycle (path) is an edge connecting two non-consecutive vertices of the cycle (path). A chordless cycle (path) is an induced subgraph that is isomorphic to a cycle (path). A graph is chordal if it contains no chordless cycle of length at least 4. A graph is an interval graph if intervals can be associated to its vertices such that two vertices are adjacent if and only if their corresponding intervals overlap. Three non-adjacent vertices form an asteroidal triple (AT) if there is a path between every two of them that does not contain a neighbor of the third. A graph is AT-free if it contains no AT. A graph is an interval graph 3

4 if and only if it is chordal and AT-free [21]. A vertex set S V is called dominating if every vertex not contained in S is adjacent to some vertex in S. A pair of vertices {u, v} is called a dominating pair if every u, v-path is dominating. Every interval graph has a dominating pair [3], and thus also a dominating chordless path. A clique tree of a graph G is a tree T whose nodes (also called bags) are maximal cliques of G such that for every vertex v in G, the subtree T v of T that is induced by the bags that contain v is connected. A graph is chordal if and only if it has a clique tree [1]. A clique path Q of a graph G is a clique tree that is a path. A graph G is an interval graph if and only if has a clique-path [10]. An interval graph has at most n maximal cliques. Given two vertices u and v in G, a vertex set S is a u, v-separator if u and v belong to different connected components of G S. A u, v-separator S is minimal if no proper subset of S is a u, v-separator. S is a minimal separator of G if there exist two vertices u and v in G with S a minimal u, v-separator. For a chordal graph G, a set of vertices S is a minimal separator of G if and only if S is the intersection of two neighboring bags in any clique tree of G [1]. An interval supergraph H = (V, E F ) of a given graph G = (V, E), with E F =, is called an interval completion of G. H is called a k-interval completion of G if F k. The set F is called the set of fill edges of H. On input G and k, the k-interval Completion problem asks whether there is an interval completion of G with at most k fill edges. In the following sections, some proofs are moved to an appendix at the end of the paper. These results are marked with an asterisk. 3 Non-chordality and small simple AT-witnesses: Rules 1, 2 Branching Rule 1: If G not chordal, find a chordless cycle C of length at least 4. If C > k+3 answer no, otherwise: Branch on the at most 4 C different ways to add an inclusion minimal set of edges (of cardinality C 3) between the vertices of C to make C chordless. If Rule 1 applies we branch by creating at most 4 C recursive calls, each with new parameter value k ( C 3). The correctness of Rule 1 is well understood [17, 2]. Let us remark that each invocation of the recursive search tree subroutine will apply only one of four branching rules. Thus, if Rule 1 applies we apply it and branch, else if Rule 2 applies we apply it and branch, else if Rule 3 applies we apply it and branch, else apply Rule 4. Rules 2, 3 and 4 will branch on single fill edges, dropping the parameter by one. Also Rule 1 could have branched on single fill edges, simply by taking the set of non-edges of the induced cycle and branching on each non-edge separately. We continue with Rule 2. Observation 3.1 Given a graph G, let {a, b, c} be an AT in G. Let P ab be the set of vertices on a path between a and b in G N[c], let P ac be the set of vertices on a path between a and c in G N[b], and let P bc be the set of vertices on a path between b and c in G N[a]. Then any interval completion of G contains at least one fill edge from the set {cx x P ab } {ax x P bc } {bx x P ac}. 4

5 Proof. Otherwise {a, b, c} would still be an independent set of vertices with a path between any two avoiding the neighborhood of the third, in other words it would be an AT. We introduce simple AT-witnesses and give a branching rule for small such witnesses. Definition 3.2 Let a, b, c be three vertices of a graph G. We define P ab to be the set of vertices on a shortest path between a and b in G N[c], P ac the set of vertices on a shortest path between a and c in G N[b], and P bc the set of vertices on a shortest path between b and c in G N[a]. Note that the three paths exist if and only if {a, b, c} is an AT. We define G abc to be the subgraph of G induced by the vertices of P ab P bc P ac, and call it a simple AT-witness for this AT. Branching Rule 2: If G is chordal: For each triple {a, b, c} check if {a, b, c} is an AT. For each AT {a, b, c}, find a simple AT-witness G abc for it. If there exists an AT {a, b, c}, such that {cx x P ab } {ax x P bc } {bx x P ac } k + 15 for the simple AT-witness G abc, then: Branch on each of the fill edges in the set {cx x P ab } {ax x P bc } {bx x P ac }. Proof. A simple AT-witness can be found in polynomial time: for a triple of vertices, check if there exists a shortest path between any two of them that avoids the neighborhood of the third vertex. Since shortest paths are used to define simple AT-witnesses, then {cx x P ab } {ax x P bc } {bx x P ac } will be the same for all simple AT-witnesses for an AT {a, b, c}. By Observation 3.1, any interval completion contains at least one edge from the set branched on by Rule 2. Lemma 3.3 Let G be a graph to which Rule 1 cannot be applied (i.e. G is chordal). There exists a polynomial time algorithm that finds a simple AT-witness G abc, where {cx x P ab } {ax x P bc } {bx x P ac } k + 15, if such an AT-witness exists. 4 Thick AT-witnesses and shallow vertices: Rule 3 In this section we introduce minimal simple AT-witnesses and show that they each have a shallow vertex. We then introduce thick AT-witnesses showing that also minimal thick AT-witnesses have a shallow vertex, and give a branching rule based on this. In Subsection 4.1 we consider graphs to which Rule 1 cannot be applied (chordal graphs) and in Subsection 4.2 graphs to which neither Rule 1 nor Rule 2 can be applied. 4.1 G is a chordal graph Observation 4.1 * Let G abc be a simple AT-witness in a chordal graph G. Then a, b, c are simplicial vertices in G abc. Definition 4.2 A simple AT-witness G abc is minimal if G abc x is AT-free for any x {a, b, c}. Observation 4.3 * Let G abc be a minimal simple AT-witness in a chordal graph. Then for any x {a, b, c}, G abc x is an interval graph, where {a, b, c} \ {x} is a dominating pair. 5

6 Definition 4.4 Let {a, b, c} be an AT in a chordal graph G. P ac 4 and P bc 4. Vertex c is called shallow if Lemma 4.5 Let G abc be a minimal simple AT-witness in a chordal graph, where P ab P bc P ac. If P ab 6 then c is shallow. Proof. Let us on the contrary assume that P bc 5. Let a = v 1, v 2,...v r = b be the path P ab, and let c be the neighbor of c in the path P bc. Notice that, by Observation 4.3 G abc a is an interval graph, where {b, c} is a dominating pair, and thus v 2 is adjacent to at least one vertex in any b, c-path in G abc a, and c is adjacent to at least one vertex in any a, b-path in G abc c. Let i be the smallest integer such that v i is a neighbor of c. We can now conclude that i 3, since v 2 is adjacent to some vertex on the path c, c, v i, v i+1,..., v r, b. Vertex set {a, v r 1, c} is an independent set; av r 1 E since r 6; cv r 1 E by definition of P ab ; and ac E since a, b, c is an independent set. The path a, v 2,...v r 1 avoids N(c) by definition of P ab ; either path a, v 2, c, c or path a, v 2, v 3, c, c avoids N(v r 1 ), since r 6 and since the shortest b, c-path in G abc a contains at least five vertices ( P bc 5); and P bc b + v r 1 is a path that avoids N(a), since r 6 and b is by Observation 4.1 a simplicial vertex in G abc. The simple witness for the AT {a, v r 1, c} induced by the paths (a, v 2,...v r 1 ), (a, v 2, c, c) or (a, v 2, v 3, c, c), and P bc b + v r 1 is now a contradiction to G abc being a minimal simple AT-witness. Observation 4.6 * A vertex v is simplicial only if v is an end vertex of every chordless path that contains v. Definition 4.7 Let {a, b, c} be an AT in a chordal graph G, and let W = {w w is a vertex of a chordless a, b-path, a, c-path, or b, c-path in G}. The graph G T abc = G[W ] is a thick AT-witness for the AT {a, b, c}. We denote the neighborhoods of a, b, and c in G T abc by respectively S a, S b, and S c, since these are minimal separators in G T abc and also in G by the following two observations. Observation 4.8 * Let G T abc be a thick AT-witness in a chordal graph G. For any x {a, b, c}, x is a simplicial vertex and S x = N GT abc (x) is a minimal separator in G T abc. Observation 4.9 * Let G T abc be a thick AT-witness in a chordal graph G. Then the set of minimal separators of G T abc are exactly the set of minimal a, b-separators, a, c-separators, and b, c-separators of G. Definition 4.10 A thick AT-witness G T abc is minimal if G T abc x is AT-free for every x {a, b, c}. Observation 4.11 * Let G T abc be a minimal thick AT-witness in the chordal graph G. Then G T abc c is an interval graph, where {a, b} is a dominating pair. 6

7 4.2 G is chordal and Rule 2 does not apply Lemma 4.12 Let G T abc be a minimal thick AT-witness in a graph G to which neither Rule 1 (i.e. G chordal) nor Rule 2 can be applied. Then at least one of the vertices in the AT {a, b, c} is shallow, and there exists a minimal simple AT-witness G abc, where V (G abc ) V (G T abc ). Proof. Let P ab, P ac, P bc be shortest chordless paths contained in G T abc, and let G abc be defined by P ab, P ac, P bc. It is clear that G T abc is minimal only if G abc is minimal. By Lemma 4.5, Rule 2, and the fact that G abc is a minimal AT-witness, we know that at least one of the vertices in {a, b, c} are shallow. Lemma 4.13 * Let G be a graph to which neither Rule 1 nor Rule 2 can be applied, and let G T abc be a minimal thick AT-witness in G where c is shallow. Then every vertex in S c is adjacent to every vertex in S a S b. Lemma 4.14 Let G = (V, E) be graph to which neither Rule 1 nor Rule 2 can be applied. Let G T abc be a minimal thick AT-witness in G where c is shallow. Let C c be the connected component of G S c that contains c. Then every vertex of C c has in G the same set of neighbors S c outside C c, in other words u C c : N G (u) \ C c = S c. Proof. By definition N G (u) \ C c S c. Let us assume for a contradiction that ux E for some x S c and u C c. Since C c is a connected component there exists a path from u to c inside C c. Let u, c be two consecutive vertices on this path, such that S c N G (c ) and u x E for some x S c. This is a contradiction, since by Lemma 4.13 x creates a short path from a to b that avoids the neighborhood of u, and by using P ac c and P bc c and the vertices c and u we can create short paths from a to u and from b to u that avoid the neighborhoods of b and a. This is now a contradiction, since {a, b, u } is an AT with a simple AT-witness where the number of branching fill edges are 5 for the path a, a, x, b, b, 5 for P ac c and c, u, and 5 for P bc c and c, u, giving a total of 15 branching edges. Lemma 4.15 Let G be graph to which neither Rule 1 nor Rule 2 can be applied, and let G T abc be a thick AT-witness in G. Then there exists a minimal thick AT-witness G T xyz in G, where V (G T xyz ) V (G T abc ) and z is shallow, such that z {a, b, c}. Proof. G T xyz will be obtained from G T abc by deleting one of the simplicial vertices in the AT that defines G T abc, and repeat this until a minimal thick AT-witness G T xyz is obtained. Note that only neighbors of the deleted vertex can become simplicial after each deletion, by Observation 4.6. As a result, the deleted vertices induce at most three connected components, where each of the components is adjacent to one of the vertices x, y, z. By Lemma 4.12 one of the vertices x, y, z is shallow. Let us without loss of generality assume that z is the shallow vertex in G T xyz. By Lemma 4.9, minimal separators of G T xyz are also minimal separators of G T abc, so let us assume without loss of generality that z and c are contained in the same connected component of G T abc N GT xyz (z). Notice that z and c might be the same vertex. By Lemma 4.14, c is shallow in the minimal thick AT-witness G T xyc. 7

8 Definition 4.16 Given a graph G to which Rules 1 and 2 do not apply we compute a set C(G) = C 1 C 2... C r of vertices that are shallow in some minimal thick AT-witness, with G \ C(G) = R r an interval graph, as follows: R 0 := G; i := 0; C(G) := ; while R i is not an interval graph do i := i + 1; Find G T ai b i c i a minimal thick AT-witness in R i 1 with c i shallow; Let C i be the connected component of R i 1 N GT ai b i c i (c i ) that contains c i ; for each c C i do G T ai b i c := G T ai b i c i c i + c; R i := R i 1 C i ; C(G) := C(G) C i ; end-while r := i The minimal thick AT-witness G T ai b i c i is found by first finding an AT {a, b, c}, then removing simplicial vertices different from a, b, c according to Observation 4.6 to get a thick AT-witness, and then applying the procedure in the proof of Lemma Note that we also computed graphs G = R 0 R 1... R r, with R r interval, and a minimal thick AT-witness for each c C(G) (from the thick minimal AT-witness G T ai b i c i with c i C i, we defined, for any c C i, the graph G T ai b i c := G T ai b i c i c i + c, which will be a thick minimal AT-witness for {a i, b i, c} with c shallow by Lemma 4.14) that will be used in the next section. First we give Branching Rule 3. Branching Rule 3: This rule applies if Rules 1 and 2 do not apply and C(G) > k, in which case we let B be a subset of C(G) with B = k + 1. For each c B find a simple AT-witness G abc where c is shallow, with shortest paths P bc and P ac avoiding N(a) and N(b), respectively. For each c B, branch on the at most 8 fill edges {ax x P bc } {bx x P ac }. Branch on the at most B ( B 1)/2 possible fill edges {uv u, v B and uv E}. Observe that Rule 3 only needs a subset of C(G) of size k + 1, and thus an algorithm can stop the computation of C(G) when this size is reached. Lemma 4.17 * Any k-interval completion of a graph G to which neither Rule 1 or 2 can be applied contains a fill edge which is branched on by Rule 3. 5 More branching and a greedy completion: Rule 4 In this section we present the fourth and final rule and prove correctness of the resulting search tree algorithm. We now consider graphs G to which none of the Rules 1, 2, or 3 can be applied. This means that G is chordal (Rule 1), that C(G) k (Rule 3), implying that (the connected components of) G[C(G)] is an interval graph (Rule 2). Like Rules 2 and 3, Rule 4 will branch on single fill edges, but it will also consider minimal separators, based on the following two basic observations. 8

9 Observation 5.1 * If G has a minimal thick AT-witness G T abc in which P ac, P bc are shortest paths avoiding N(b) and N(a) respectively, then any interval completion of G either contains a fill edge from the set {bx x P ac } {ax x P bc } or contains one of the edge sets {{cx x S} S is a minimal a, b-separator in G T abc }. Observation 5.2 * Let G be a graph to which neither Rule 1 nor 2 can be applied, and let G T abc be a minimal thick AT-witness in G where c is shallow. Then S c S for every minimal a, b-separator S different from S a and S b. Recall that C(G) = C 1 C 2... C r was computed in Definition 4.16 by removing from G the vertex sets C i in the order from i = 1 to r. A priori we have no guarantee that there are no edges between a vertex in C i and a vertex in C j, for some i j, but when C(G) k this indeed holds, as shown in the following lemma. Lemma 5.3 * Let G = (V, E) be a graph to which none of Rules 1, 2, 3 can be applied, and let C(G) = C 1 C 2... C r from Definition Then C i induces an interval graph that is a connected component of G[C(G)], for each 1 i r. Rule 4 will branch on a bounded number of single fill edges and it will also compute a greedy completion by choosing for each shallow vertex a minimal separator minimizing fill and making the shallow vertex adjacent to all vertices of that separator. We will prove that if none of the single fill edges branched on in Rule 4 are present in any k-interval completion, then the greedy completion gives an interval completion with the minimum number of edges. The greedy choices of separators are made as follows: Definition 5.4 Let G be a graph to which none of Rules 1, 2, 3 can be applied. Let Definition 4.16 give C(G) = C 1 C 2... C r, representative vertices c 1, c 2,..., c r and minimal thick AT-witnesses G T ai b i c i and graphs G = R 0 R 1... R r, with R r interval. We compute fill-minimizing minimal separators M 1 to M r as follows: for i := 1 to r do M i := null; for each minimal a i, b i -separator S in G T ai b i c i do if S C(G) = and S S ai and S S bi and S N(C j ) for all 1 j r then if M i = null or S \ N(C i ) < M i \ N(C i ) then M i := S; end-for Lemma 5.5 * If M i null then M i is a minimal separator in R r for any 1 i r. Branching Rule 4: Rule 4 applies if none of Rules 1, 2, 3 apply, in which case we compute, as in Definitions 4.16 and 5.4, C 1, C 2,..., C r (which are connected components of G[C(G)] by Lemma 5.3), the minimal thick AT-witnesses G T ai b i c with c shallow for each c C i, and M 1,..., M r (which are minimal separators of R r by Lemma 5.5). For each 1 i r and each c C i choose a i S a i \ S c and b i S b i \ S c and find P ai c and P bi c (shortest paths in G T ai b i c avoiding N(b i ) and N(a i ), respectively, of length at most 4 by Lemma 4.12). For each pair 1 i j r, choose a vertex v i,j N(C j ) \ N(C i ) (if it exists). 9

10 For 1 i r and c C i, branch on the at most 8 fill edges {a i x x P bi c} {b i x x P ai c} and also on the 2 fill edges {ca i, cb i }. Branch on the at most C(G) ( C(G) 1)/2 fill edges {uv u, v C(G) and uv E}. Branch on the at most C(G) r fill edges 1 i j r {cv i,j c C i }. Finally, compute H = (V, E 1 i r {cx c C i and x M i }) and check if it is a k-interval completion of G (note that we do not branch on H.) Lemma 5.6 If G has a k-interval completion, and Rules 1, 2, and 3 do not apply to G, and no k-interval completion of G contains any single fill edge branched on by Rule 4, then the graph H, which Rule 4 obtains by adding fill edges from every vertex in C i to every vertex in M i for every 1 i r, is a k-interval completion of G. Proof. By Observation 5.1, for each c C i either one of the edges in {a i x x P bi c} {b i x x P ai c} is a fill edge (and all these are branched on by Rule 4) or else the k-interval completion contains the edge set {cx x S} for some minimal a i, b i -separator S in G T ai b i c. Such an edge set in a k-interval completion is one of four types depending on the separator S used to define it. For each type and any c C i we argue that Rule 4 considers it. Observe that N(C i ) \ C i = N(c) \ C i by Lemma 4.14, and thus the fill edges from c will go to vertices in S \ N(C i ), which is nonempty since there is an a i, b i -path avoiding N(c). We now give the four types of minimal separators S, and show that the first three are branched on by a single fill edge: 1. S C(G). Since N(C i ) C(G) = by Lemma 5.3, we have in this case a fill edge between two vertices in C(G) (between c C i and a vertex in C(G) S \ N(C i )) and all these are branched on by Rule S = S ai or S = S bi, where S ai, S bi, S c defined by G T ai b i c. We found in Rule 4 a pair of vertices a i S a i \ S c and b i S b i \ S c and branched on the fill edges ca i and cb i. 3. S = N(C j ) for some 1 j r. If S = N(C j ) then N(C j ) \ N(C i ) and we found in Rule 4 a vertex v i,j N(C j ) \ N(C i ) and branched on the fill edge cv i,j. 4. S is neither of the three types above. Note that M i was chosen in Definition 5.4 by looping over all minimal a i, b i -separators S in G T ai b i c i (which by Lemma 4.14 are exactly the minimal a i, b i -separators of G T ai b i c) satisfying S C(G) =, S S a, S S b, and S N(C j ) for any j. Thus, of all separators of this fourth type, M i is the one minimizing the fill. The assumption is that G has a k-interval completion but no single edge branched on by Rule 4 is present in any k-interval completion. This means that only separators of the fourth type are used in any k-interval completion. Since H added the minimum possible number of fill edges while using only separators of the fourth type any interval completion of G must add at least E(H) \ E(G) edges. It remains to show that H is an interval graph. H is constructed from an interval graph R r and the components G[C 1 ],..., G[C r ] of G[C(G)], which are interval 10

11 graphs by Lemma 5.3, and M 1,..., M r which are minimal separators of R r by Lemma 5.5. Since M i S ai and M i S bi we have by Observation 5.2 that S c = N(C i ) M i so that adding all edges between C i and M i for 1 i r gives the graph H. We show that H is an interval graph by induction on 0 i r. Let H 0 = R r and let H i for i 1 be the graph we get from H i 1 and C i by making all vertices of C i adjacent to all vertices of the minimal separator M i of R r. H 0 is an interval graph by induction, and its minimal separators include all minimal separators of R r. If (K 1, K 2,...K q ) is a clique path of H i 1 with M i = K j K j+1, and (K 1, K 2,..., K p) is a clique path of G[C i ] then (K 1, K 2,..., K j, K 1 M i, K 2 M i,..., K p M i, K j+1,..., K q ) is a clique path of H i, and hence H i is an interval graph. Finally, observe that the minimal separators of H i 1 and hence of R r are also minimal separators of H i. Theorem 5.7 The search tree algorithm applying Rules 1, 2, 3, 4 in that order decides in O(k 2k n 3 m) time whether an input graph G on n vertices and m edges can be completed into an interval graph by adding at most k edges. Proof. At least one of the rules will apply to any graph which is not interval. The correctness of Rule 1 is well understood [17, 2], that of Rules 2 and 3 follow by Observations 3.1 and 5.1 and of Rule 4 by Lemma 5.6. Each branching of Rules 2, 3 and 4 add a single fill edge and drops k by one. As already mentioned, also Rule 1 could have added a single fill edge in each of its then at most k 2 branchings. The height of the tree is thus no more than k, before k reaches 0 and we can answer no. If an interval graph is found we answer yes. Let us argue for the runtime. The graph we are working on never has more than m+k edges. In Rule 1 we decide in linear time if the graph has a large induced cycle. In Rule 2 we may have to try all triples when searching for an AT with a small simple AT-witness, taking O(n 3 (m + k)) time. In Rule 3 and 4 we need to find a minimal thick AT-witness at most k + 1 times. As observed earlier, the minimal thick AT-witness is found by first finding an AT {a, b, c}, which can be done in time O(m + k) since G is a chordal graph [19], then remove simplicial vertices different from a, b, c to find the thick AT-witness, and then make it minimal. Using a clique tree we find in this way a single minimal thick AT-witness in time O(n 3 ) and at most k of them in time O(n 3 k). Hence each rule takes time at most O(n 3 (m + k)) and has branching factor at most k 2 (e.g. in Rule 1 and also in Rule 3 when branching on all fill edges between pairs of shallow vertices). The height of the search tree is at most k and the number of nodes therefore at most k 2k. We can assume k n m since otherwise a brute-force algorithm easily solves minimum interval completion in n 2n steps. Thus each rule takes time O(n 3 m) for total runtime O(k 2k n 3 m). 11

12 References [1] P. Buneman. A characterization of rigid circuit graphs. Discrete Math., 9: , [2] L. Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58(4): , [3] D. G. Corneil, S. Olariu, and L. Stewart. Asteroidal Triple-Free Graphs. SIAM J. Discrete Math., 10(3): , [4] J. Díaz, A.M. Gibbons, M.S. Paterson, and J. Torán. The minsumcut problem. In Proceedings WADS 1991, [5] G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25:71 76, [6] R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer Verlag, New York, [7] G. Even, J. Naor, S. Rao, and B. Scheiber. Divide-and-conquer approximation algoeirhtms via spreading metrics. In Proceedings FOCS 1995, [8] M. Garey and D. Johnson. Computers and intractability: a guide to the theory of NPcompleteness W.H.Freeman and co, San Fransisco, [9] J. A. George and J. W. H. Liu. Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall Inc., Englewood Cliffs, New Jersey, [10] P. C. Gilmore and A. J. Hoffman. A characterization of comparability graphs and of interval graphs. Canadian Journal of Mathematics, 16: , [11] P. W. Goldberg, M. C. Golumbic, H. Kaplan, and R. Shamir. Four strikes against physical mapping of DNA. J. Comput. Bio., 2(1): , [12] M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Second edition. Annals of Discrete Mathematics 57. Elsevier, [13] G. Gutin, S. Szeider, and A. Yeo. Fixed-parameter complexity of minimum profile problems. In Proceedings IWPEC 2006, pages 60 71, Springer LNCS [14] P. Heggernes, J. A. Telle, and Y. Villanger. Computing Minimal Triangulations in Time O(n α logn) = o(n ). SIAM Journal on Discrete Mathematics, 19-4: , In Proceedings SODA 2005, pages , [15] P. Heggernes, K. Suchan, I. Todinca, and Y. Villanger. Minimal interval completions. In Proceedings ESA 2005, pages , Springer LNCS [16] H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping. In Proceedings FOCS 1994,

13 [17] H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput., 28(5): , [18] T. Kashiwabara and T. Fujisawa. An NP-complete problem on interval graphs. IEEE Symp. of Circuits and Systems, pages 82 83, [19] D. Kratsch, R. McConnell, K. Mehlhorn, and J. Spinrad. Certifying algorithms for recognizing interval graphs and permutation graphs In Proceedings SODA 2003, [20] D. Kuo, and G.J. Chang. The profile minimization problem in trees. SIAM J. Comput. 23, 71-81, 1994 [21] C. G. Lekkerkerker and J. C. Boland. Representation of a finite graph by a set of intervals on the real line. Fund. Math., 51:45 64, [22] S. Rao and A. Richa. New approximation techniques for some linear ordering problems. SIAM J. Comput. 34(2): (2004) [23] R. Ravi, A. Agrawal, and P. Klein. Ordering problems approximated: single processor scheduling and interval graph completion. In Proceedings ICALP , 1991 [24] M. Serna and D. Thilikos. Parameterized complexity for graph layout problems. Bulletin of the EATCS, 86:41 65, [25] R. E. Tarjan. Graph theory and Gaussian elimination. In Sparse Matrix Computations (eds. J. R. Bunch and D. J. Rose). Academic Press,

14 Appendix: Omitted proofs Proof of Observation 4.1. By the definition of G abc, N(a) 2. As P bc avoids N(a) and as any vertex of N(a) has a neighbor in the connected component of G abc N(a), N(a) is a minimal a, c-separator. G abc is a chordal graph(since it is an induced subgraph of a chordal graph), and by [5] every minimal separator of a chordal graph is a clique, thus N(a) is a clique. Proof of Observation 4.3. We prove the observation for x = c, and the other two possibilities are symmetric. Clearly, G = G abc c is an interval graph, since G is chordal and G abc is a minimal simple AT-witness. For a contradiction assume that {a, b} is not a dominating pair in G ; thus there exists a path P ab from a to b in G N[y] for some vertex y V (G ) \ {a, b}. Let Q be a clique path of G. Vertex y does not appear in any bag of Q that contains a or b, and it does not appear in any bag between subpaths Q a and Q b of Q. Let us without loss of generality assume that Q a appears between Q y and Q b in Q. Because of this y is not contained in the component C b of G N[a] that contains b. Furthermore, a is a simplicial vertex by Observation 4.1, and P ab contains vertices from N(a), thus y N(a) since P ab would not avoid the neighborhood of y otherwise. The path P bc c is contained in C b, and thus y is not adjacent to any vertex in P bc c. We know that cy E, since by Observation 4.1 N(c) is a clique, and thus y would be adjacent to the neighbor of c in P ab if cy were an edge. None of the paths P ab, P ac, P bc contains y, since P bc is strictly contained in C bc, and since any shortest path from a to either b or c only contains one neighbor of a. Thus, G abc is not a simple AT-witness for {a, b, c}, since it contains y. Proof of Observation 4.6. Any vertex that appears as a non end vertex in a chordless path, has two neighbors that are not adjacent. Proof of Observation 4.8. We prove the observation for x = a; the other possibilities are symmetric. Every neighbor of a in G T abc appears in a chordless path from a to either b or c or both. Because of the existence of the path P bc avoiding S a, it follows that S a is a minimal separator. In a chordal graph, every minimal separator is a clique [5]. Hence a is simplicial in G T abc. Proof of Observation 4.9. Every minimal separator of G T abc separates two simplicial vertices appearing in two different leaf bags of any clique tree of G T abc. Since a, b, c are the only simplicial vertices in G T abc, every minimal separator of G T abc is a minimal a, b-separator, b, c-separator, or a, c-separator. Let S be a minimal a, b-separator in G. Then there exist two connected components C a and C b of G S, containing respectively a and b, such that N G (C a ) = N G (C b ) = S. For any vertex z S we can now find a chordless shortest path in G from z to each of a and b, where every intermediate vertex is contained in respectively C a and C b. By joining these two paths, we get a chordless path from a to b that contains z. Since this holds for any vertex in S, it follows by the way we defined G T abc that any minimal a, b-separator of G is a minimal a, b-separator of G T abc. The argument can be repeated with a, c and b, c to show that every minimal a, c-separator or b, c-separator of G is also a minimal separator of G T abc. Let S be a minimal a, b-separator in G T abc Vertex set S is a subset of a minimal a, b-separator of G, since the same chordless paths exist in G. But S cannot be a proper subset of a minimal 14

15 a, b-separator of G, since every minimal a, b-separator of G is a minimal a, b-separator in G T abc, and thus S would not be a minimal separator in G T abc otherwise. The argument can be repeated with a, c and b, c. Proof of Observation The graph G = G T abc c is by definition an interval graph, since G T abc is a minimal thick AT-witness. For a contradiction assume that {a, b} is not a dominating pair, and thus there exists a path P ab from a to b in G N[y] for some vertex y V (G ) \ {a, b}. Let Q be a clique path of G. Vertex y does not appear in any bag of Q that contains a or b, and it does not appear in any bags between the subpaths Q a and Q b of Q. Let us without loss of generality assume that Q a appears between Q y and Q b in Q. We show that y is then not in any chordless path between any pair of a, b, c, giving the contradiction. Due to the above assumptions, y is not contained in the component C b of G N[a] that contains b. Furthermore, a is a simplicial vertex by Observation 4.8, and P ab contains vertices from N G (a), thus y N G (a) since P ab would not avoid the neighborhood of y otherwise. The path P bc c is contained in C b since it contains no vertex of N[a], and thus y is not adjacent to any vertex in P bc c. We know that cy E(G T abc ), since by Observation 4.8, N GT abc (c) is a clique, and thus y would be adjacent to the neighbor of c in P ab if cy were an edge in E(G T abc ). Now we have a contradiction since y is not in any chordless path between any pair of a, b, c. Proof of Lemma Let E = E(G T abc ), and let us on the contrary and without loss of generality assume that c a E for c S c and a S a. Let P ab = (a = v 1, v 2,..., v r = b), P bc, and P ac be the shortest paths used to define a simple AT-witness for {a, b, c}. We will show that either {a, b, c} or {a, v r 1, c} is an AT in a subgraph of G T abc, contradicting its minimality. Vertex set {a, b, c} is an independent set since cb E, a b E due to P ab > 15 8 (Rule 2), and a c E because c is simplicial in G T abc, and thus c a E if a c E. Either v 2 = a, or a v 2 E since a is simplicial in G T abc. P ab a + a is a path from a to b that avoids the neighborhood of c. In the same way P ac a + a is a path from a to c, and since P ab > 7 this path avoids the neighborhood of b. By Observation 4.11, c is adjacent to some vertex on the path P ab = (a = v 1, v 2,..., v r = b). If c is adjacent to some vertex v i where i > 3, then there is a path c, c, v i,..., v r = b that avoids the neighborhood of a, and we have a contradiction since a, b, c would be an AT in G T abc a. We can therefore assume that v j c E, where j {2, 3}, and that there exists no v i c E for any i > 3. The set {a, v r 1, c} is an independent set, since cv r 1, av r 1 E. The path a, v 2,..., v r 1 avoids the neighborhood of c, the path c, c, v j,..., a avoids the neighborhood of v r 1, and P bc b + v r 1 is a path from c to v r 1 that avoids the neighborhood of a, since b is simplicial in G T abc. This is a contradiction since G T abc b contains the AT {a, v r 1, c}. Proof of Lemma In a k-interval completion we cannot add more than k fill edges. Thus, since B = k+1 any k-interval completion H of G either contains a fill edge between two vertices in B (and all these are branched on by Rule 3), or there exists a vertex c B with no fill edge incident to it (since the opposite would require k + 1 fill edges). If c B does not have a fill edge incident to it, then by Observation 3.1 one of the edges in {ax x P bc } {bx x P ac } must be a fill edge (and all these are branched on for each c B by Rule 3). Proof of Observation 5.1. By Observation 3.1, we know that at least one of the edges in {ax x P bc } {bx x P ac } {cx x P ab } for the paths P ab, P ac, P bc defined in the proof of 15

16 Lemma 4.12, is a fill edge of any interval completion of G. If an interval completion H does not contain any fill edge from the set {bx x P ac } {ax x P bc }, then H contains at least one fill edge from the set {cx x P ab }, where P ab is any chordless a, b-path in G that avoids the neighborhood of c. Thus, N H (c) contains a minimal a, b-separator in G (which by Observation 4.9 is also a minimal a, b-separator in G T abc ) since every chordless and thus every a, b-path in G N[c] contains a vertex of N H (c). Proof of Observation 5.2. Let S be a minimal a, b-separator different from S a and S b. S is then also a minimal a, b -separator for some a S a and some b S b, since no minimal a, b-separator contains another minimal a, b-separator as a subset. It then follows from Lemma 4.13 that S c N(a ) N(b ), and thus S c S. Proof of Lemma 5.3. Firstly, since C i k and Rules 1, 2 do not apply, it must induce an interval graph. To argue that it is a connected component, note first that by definition G[C i ] is connected and C i C j = for any i j. For a contradiction we assume that cz E for some c C i and z C j with i < j. Let G T abc be the minimal thick AT-witness in R i 1 with c the shallow vertex and S c = N GT abc (c), and let likewise G T xyz be the minimal thick AT-witness in R j 1 with z shallow and S z = N GT xyz (z). Let P ab be a path from a to b in G T abc \ N(c). There are now two cases: Case I: There is a vertex w P ab S z. By Observation 4.9 both S c and S z are minimal separators in the chordal graph G, and thus S c, S z are cliques [5]. Thus, since cw E we must have c S z. But then we have c and z in the same component C z of G \ S z. By Lemma 4.14 c and z must therefore have the same neighbors outside C z. But this contradicts the fact that zw E while cw E. Case II: P ab S z =. Let C z be the connected component of G \ S z that contains z. By Lemma 4.13 we have zw E for some w P ab and therefore V (P ab ) C z. By Lemma 4.5 and the fact that Rule 2 cannot be applied we have at least k vertices in P ab and thus C z P ab > k. Assuming we can show the subset-property C z C 1 C 2... C j we are done with the proof since this will lead to the contradiction k < C z C 1 C 2... C j C(G) k. Let us prove the subset-property. G has a perfect elimination ordering starting with the vertices of C 1, as these vertices are a component resulting from removing a minimal separator from G. By induction, we have that G has a perfect elimination ordering α starting with the vertices in C 1 C 2... C j 1. For a contradiction assume there exists a vertex w C z \(C 1 C 2... C j ). As w C z there is a shortest w, z-path P wz in C z. Since zw E, P wz contains at least 3 vertices and one of these vertices belongs to some C i, if not w would belong to C j. Let s be the first vertex in the ordering α that belongs to the path P wz. This is now a contradiction since a none end vertex of a chordless path cannot be simplicial. Proof of Lemma 5.5. The vertex set M i is a minimal separator in G T ai b i c i by construction and since G T ai b i c i is a subgraph of the chordal graph R i it is by Observation 4.9 also a minimal separator of R i. We prove that M i is also a minimal separator in R j for any i + 1 j r by induction on j. Recall that R j is obtained by removing C j from R j 1, where C j is a component of R j 1 \ S ci for a minimal separator S ci of R j 1, and S ci = N(C j ) by Lemma Consider a clique tree of R j 1 and observe that any minimal separator of R j 1 that is not a minimal separator of R j is either equal to N(C j ) or it contains a vertex of C j. Finally, note that the minimal separator M i has been chosen so that it is not of this type. 16